To solve this problem, we need to use the mirror formula for a concave mirror. The mirror formula states:
[tex]\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \][/tex]
where:
- [tex]\( f \)[/tex] is the focal length of the mirror,
- [tex]\( v \)[/tex] is the image distance from the mirror,
- [tex]\( u \)[/tex] is the object distance from the mirror.
Given:
- The focal length [tex]\( f \)[/tex] is 30 cm,
- The object distance [tex]\( u \)[/tex] is 50 cm.
It's important to note that for concave mirrors, the object distance [tex]\( u \)[/tex] is taken as negative if the object is placed in front of the mirror, so [tex]\( u = -50 \)[/tex] cm.
Let's plug these values into the mirror formula and solve for [tex]\( v \)[/tex].
1. Write down the mirror formula:
[tex]\[
\frac{1}{f} = \frac{1}{v} + \frac{1}{u}
\][/tex]
2. Substitute the given values [tex]\( f = 30 \)[/tex] cm and [tex]\( u = -50 \)[/tex] cm into the formula:
[tex]\[
\frac{1}{30} = \frac{1}{v} + \frac{1}{-50}
\][/tex]
3. Rewrite the equation to make it easier to solve for [tex]\( \frac{1}{v} \)[/tex]:
[tex]\[
\frac{1}{30} = \frac{1}{v} - \frac{1}{50}
\][/tex]
4. Combine the fractions on the right side:
[tex]\[
\frac{1}{30} + \frac{1}{50} = \frac{1}{v}
\][/tex]
5. Find the common denominator and combine the fractions on the left side:
[tex]\[
\frac{1}{30} + \frac{1}{50} = \frac{50 + 30}{1500} = \frac{80}{1500} = \frac{8}{150} = \frac{4}{75}
\][/tex]
6. Therefore:
[tex]\[
\frac{1}{v} = \frac{4}{75}
\][/tex]
7. To find [tex]\( v \)[/tex], take the reciprocal of both sides:
[tex]\[
v = \frac{75}{4}
\][/tex]
8. Simplify the fraction:
[tex]\[
v = 18.75 \text{ cm}
\][/tex]
So, the distance of the image from the mirror is 18.75 cm.
